AFCAT Numerical Ability · Profit and Loss

Marked Price, Discount and Successive Discounts

A discount is taken on the marked price and a profit on the cost price; successive discounts multiply, and a markup followed by a discount is one product of two multipliers.

All Profit and Loss notes

Why this matters

Two bases meet on this page: the discount is on the marked price and the profit on the cost. One AFCAT past-year question, from 2016, asks for the cost from a marked price, a discount and a profit. Most lost marks come from adding successive discounts or taking a profit on the marked price.

Concept 1 of 4: Discount on the marked price

The marked price (MP), also called the list or label price, is the price on the tag. A discount is a percentage of the MP, and the customer pays the rest.

Definition

Discount $= MP - SP$ ; discount % $= \dfrac{\text{discount}}{MP} \times 100$ .
After a $d\%$ discount, $SP = MP\left(1 - \tfrac{d}{100}\right)$ , so $MP = SP \div \left(1 - \tfrac{d}{100}\right)$ .
Two discounts compared on the same MP: the extra $d\%$ is worth $d\%$ of the MP.

Selling price after a discount

SP = MP\left(1 - \dfrac{d}{100}\right)

Worked example

A jacket marked ₹2,400 is sold for ₹1,980. Find the discount per cent. At what price would a

20\%

discount sell it?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Profit and Loss · Marked Price and Discounts

A watch was sold for ₹1,440 after giving a discount of 10%. The marked price of the watch was:

The discount is on the marked price

A jacket marked ₹2,400 sold for ₹1,980 has a discount of 420 on 2400,

17.5\%

. Dividing by 1980 gives

21.2\%

, which no shop means.

Divide to undo a discount

A price of ₹960 after a

20\%

discount comes from

MP = 960 \div 0.8 = 1200

. Adding

20\%

to 960 gives 1152.

Drill 7 more on discount on the marked price

Concept 2 of 4: Successive discounts

The second discount is taken on the price left after the first, so the multipliers multiply. Two discounts together are always worth less than their sum.

Definition

Discounts of $a\%$ then $b\%$ : the buyer pays $\left(1 - \tfrac{a}{100}\right)\left(1 - \tfrac{b}{100}\right)$ of the MP.
The single equivalent discount is $a + b - \dfrac{ab}{100}$ per cent.
The order of the discounts does not matter.
Three discounts: multiply three factors, or combine two at a time.

Single equivalent of two discounts

a + b - \dfrac{ab}{100}

Worked example

Find the single discount equivalent to successive discounts of

25\%

and

12\%

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Profit and Loss · Marked Price and Discounts

What is the single equivalent discount for successive discounts of 20% and 10%?

Discounts do not add

25\%

and

12\%

together are

34\%

, not

37\%

: the

12\%

is taken on a price already cut.

A rupee figure is a share of the marked price

If the equivalent discount comes to a rupee amount, that amount is the single percentage of the MP. Divide by the percentage, not by what the buyer paid.

Drill 8 more on successive discounts

Concept 3 of 4: Markup, then discount

Marking up by

m\%

multiplies the cost by

1 + \tfrac{m}{100}

; the discount then multiplies by

1 - \tfrac{d}{100}

. The profit is however much the product exceeds 1.

Definition

$SP = CP\left(1 + \tfrac{m}{100}\right)\left(1 - \tfrac{d}{100}\right)$ .
Profit % $= m - d - \dfrac{md}{100}$ ; a negative answer is a loss.
A markup and a discount of the same rate always lose: $20\%$ up and $20\%$ off leaves 0.96 of the cost.
To compare two discount plans on one MP, take the cost as 100 and work out each selling price.

Profit after a markup and a discount

\text{Profit}\% = m - d - \dfrac{md}{100}

Worked example

A shopkeeper marks his goods

50\%

above cost and allows a discount of

12\%

. Find his profit per cent.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Profit and Loss · Marked Price and Discounts

A trader marks his goods at 40% above the cost price and gives a discount of 10%. What is his profit percentage?

Subtracting the rates overstates the profit

50\%

up and

12\%

off is a profit of

32\%

, not

38\%

: the discount is taken on the raised price.

Same rate up and down is a loss

30\%

up and

30\%

off gives

1.3 \times 0.7 = 0.91

of the cost, a

9\%

loss, not 'no profit, no loss'.

Drill 15 more on markup, then discount

Concept 4 of 4: Marked price for a target profit

Work backwards from the cost. The target profit gives the selling price, and the discount says what fraction of the marked price that selling price is.

Definition

$MP = CP \times \dfrac{100 + p}{100 - d}$ .
The markup needed is $\dfrac{100 + p}{100 - d} - 1$ .
From the MP back to the cost: $CP = MP \times \dfrac{100 - d}{100 + p}$ .
Discount needed for a target profit: get the SP from the cost, then the discount is $\dfrac{MP - SP}{MP}$ .
A retailer who buys at a discount on the list price and sells at a smaller discount: his CP and SP are both shares of the list price.

Marked price for a target

MP = CP \times \dfrac{100 + p}{100 - d}

Worked example

A dealer wants a profit of

17\%

after allowing a discount of

22\%

. By what per cent above cost must he mark his goods?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2016 · AFCAT (I) 2016 — Numerical Ability · 02-212Easy

Example 4 · Profit and Loss · Marked Price and Discounts

A shopkeeper offers his customers a 10% discount and still makes a profit of 26%. What is the actual cost to him of an article marked ₹280?

Divide by the discount factor

A selling price of ₹612 after

15\%

off means

MP = 612 \div 0.85 = 720

. Multiplying by 1.15 gives 703.80.

The profit is on the cost, not on the marked price

20\%

profit is

20\%

of the cost. Taking it on the marked price gives a selling price that is too high.

Drill 10 more on marked price for a target profit

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Discount on the marked price
Selling price after a discount
$SP = MP\left(1 - \dfrac{d}{100}\right)$
Successive discounts
Single equivalent of two discounts
$a + b - \dfrac{ab}{100}$
Markup, then discount
Profit after a markup and a discount
$\text{Profit}\% = m - d - \dfrac{md}{100}$
Marked price for a target profit
Marked price for a target
$MP = CP \times \dfrac{100 + p}{100 - d}$

Watch out for (8)

The discount is on the marked price
→ Discount on the marked price
Divide to undo a discount
→ Discount on the marked price
Discounts do not add
→ Successive discounts
A rupee figure is a share of the marked price
→ Successive discounts
Subtracting the rates overstates the profit
→ Markup, then discount
Same rate up and down is a loss
→ Markup, then discount
Divide by the discount factor
→ Marked price for a target profit
The profit is on the cost, not on the marked price
→ Marked price for a target profit

Drill the questions on this topic

44 questions from the bank, with cart and Word export.

Drill all 44 questions